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equations for lepton flavor violation
Sacha Davidson, Martin Gorbahn, Matthew Leak
To cite this version:
Majorana neutrino masses in the renormalization group equations
for lepton flavor violation
Sacha Davidson*
LUPM, CNRS, Universit´e Montpellier Place Eugene Bataillon, F-34095 Montpellier, Cedex 5, France
Martin Gorbahn† and Matthew Leak‡
Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom
(Received 21 August 2018; published 20 November 2018)
We suppose that the observed neutrino masses can be parametrized by a lepton number violating dimension-five operator, and calculate the mixing of double insertions of this operator into lepton flavor changing dimension-six operators of the standard model effective theory. This allows to predict the log-enhanced, but m2ν-suppressed lepton flavor violation that is generic to high-scale Majorana neutrino mass models. We also consider the two Higgs doublet model, where the second Higgs allows the construction of three additional dimension-five operators, and evaluate the corresponding anomalous dimensions. The sensitivity of current searches for lepton flavor violation to these additional Wilson coefficients is then examined.
DOI:10.1103/PhysRevD.98.095014
I. INTRODUCTION
Neutrinos are elusive and enigmatic particles: uncolored, uncharged, and very light. Nonetheless, their observed masses and mixing angles [1] imply that lepton flavor violation (LFV) must occur, where we define LFV as flavor-changing contact interactions of charged leptons (for a review, see, e.g., [2]). Since these do not occur in the standard model (SM), LFV is considered to be “new physics”, and searched for in a wide variety of experiments
[1,3–12]. Neutrinos could also induce another kind of new physics: if their small masses are “Majorana,” they are lepton number violating (LNV), and could for instance mediate neutrinoless double-β-decay[13]. Below the weak scale, such masses appear as renormalizable terms in the Lagrangian, but in the full SU(2) gauge invariant standard model, they arise as a nonrenormalizable, dimension-five operator.
In this paper, we assume that neutrino masses are Majorana, and generated by New Physics in the lepton sector at a scale Λ ≫ mW. We focus on the theory above
mW but below Λ, where it can be described in the framework of the standard model effective field theory1 (SMEFT). The neutrino masses are parametrized by oper-ators of dimension five, and LFV is parametrized by operators of dimension-six. Our aim is to obtain the log-enhanced loop contributions of two LNV operators to LFV processes, which arise in the renormalization group equa-tions (RGEs). In particular, we calculate the anomalous dimensions that mix two dimension-five operators into a dimension-six operator. The renormalization group running of the dimension-five operators has been extensively studied in the literature [16–18], and the mixing of the dimension-six operators among themselves have been evaluated at one-loop [19]in the “Warsaw”-basis[20] of SMEFT operators. The mixing of two dimension-five operators into dimension-six operators was calculated in
[21], using the Buchmuller-Wyler[22]basis at dimension-six. We perform this calculation using the“Warsaw”-basis, and our results appear to disagree with [21].
The mixing of neutrino masses into LFV amplitudes is Oðmν=mWÞ2lnðΛ=mWÞ, so negligibly small, but completes
the anomalous dimensions required to perform a one-loop renormalization-group analysis of the SMEFT at dimen-sion-six. Indeed, this mixing does not involve any SM couplings, so in a coupling expansion, would be the leading contribution to the one-loop RGEs of the SMEFT; it is only small because the dimension five coefficients are small. In addition, we explore an extension of the SMEFT with two *[email protected]
†[email protected] ‡[email protected]
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation,
Higgs doublets[23], where the second Higgs doublet lives at a scale m22 between mW and significantly below the
lepton number/flavor-changing scaleΛ, and we impose that LFV at the weak scale is still described by the dimension-six operators of the SMEFT. In this scenario, there are four LNV dimension-five operators above m22, but only one combination of coefficients contributes to neutrino masses. We calculate the mixing of these LNV operators into the LFV operators of the SMEFT, and estimate the sensitivity of current LFV experiments to their coefficients.
The paper is organized as follows. In Sec.IIwe introduce the notation of our standard model and two-Higgs doublet model calculation. The main results are presented in Sec. III, where we discuss the general structure of our calculation and give the relevant counterterms, anoma-lous dimensions and renormalization group equations. Section IV discusses the phenomenological implications of both results before we conclude. We provide the relevant Feynman rules, further details of the calculation (including a careful treatment of the flavor structures), and the renormalization group in the Appendices A–C. The LFV operators of the SMEFT are recalled in AppendicesDandE
gives the current experimental constraints on some LFV coefficients of the SMEFT at the weak scale. AppendixF
provides a comparison with the previous calculation of[21]
and AppendixG presents the lepton conserving contribu-tions to the anomalous dimensions.
II. NOTATION AND REVIEW The SM Lagrangian for leptons can be written as Llep¼ ilαγμDμlαþ ieαγμDμeα− ð½YeαβlαHeβþ H:c:Þ
ð2:1Þ where Greek letters represent lepton generation indices in the charged-lepton mass eigenstate basis, ½Ye is the
diagonal charged-lepton Yukawa matrix with eigenvalues yα, l is a doublet of left-handed leptons, and e is a right-handed charged-lepton singlet. The explicit form of the lepton and Higgs doublets is
l ¼ νL eL ; H¼ Hþ H0 ; ð2:2Þ
which have hypercharge YðlÞ ¼ −1=2 and YðHÞ ¼ 1=2 respectively. The covariant derivative for a lepton doublet is ðDμlÞi α¼ δij∂μþ i g 2τaijWaμþ iδijg0YðlÞBμ lj α; ð2:3Þ where τa are the Pauli matrices. This sign convention for the covariant derivative agrees with[19].
Heavy new physics can be parametrized by adding nonrenormalizable operators to the SM Lagrangian that respect the SM gauge symmetries [22]. There is only a
single operator at dimension-five in the SM, which is the lepton number violating“Weinberg” operator[24]which is responsible for Majorana masses of left-handed neutrinos. The resulting effective Lagrangian at dimension-five is δL5¼C αβ 5 2ΛðlαεHÞðlcβεHÞ þ Cαβ5 2Λ ðlcβεHÞðlαεHÞ; ð2:4Þ where ε is the totally antisymmetric rank-2 Levi-Civita symbol with ε12¼ þ1, all implicit SU(2) indices inside brackets are contracted, and the charge conjugation acts on the SU(2) component li of the lepton doublet
asðliÞc¼ CliT. The charge conjugation matrix C fulfils
the properties of the charge-conjugation matrix used in
[25].2 The coefficient Cαβ5 is symmetric under the inter-change of the generation indicesα, β, the new physics scale Λ is assumed ≫ mW, and the second term is the Hermitian
conjugate of the first.
In the broken theory, with H0¼ v þ ðh=pffiffiffi2Þ, v ≃ mt,
this gives a Majorana neutrino mass matrix δL ¼ −1
2½mναβνανcβþ H:c: ½mναβ¼ −
v2
ΛCαβ5 ð2:5Þ
In the charged lepton mass eigenstate basis, this mass matrix is diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata matrix½mναβ ¼ UαimνiUβi.
At dimension-six, we will be interested in SM-gauge invariant operators that violate lepton flavor; a complete list is given in Appendix D. Following the conventions of
[19,20], they are added to the Lagrangian as: δL6¼
X
X;ζ
CζX
Λ2OζXþ H:c: ð2:6Þ
where X is an operator label andζ represents all required generation indices which are summed over all generations. Of particular interest are the operators that can be generated at one-loop with two insertions of dimension-five oper-ators, as illustrated in Fig. 1. With SM particle content, these operators involve two Higgs doublets and two lepton doublets, four lepton doublets, or three Higgs doublets and leptons of both chiralities. In the“Warsaw” basis[20], the possibilities at dimension-six are
OαβHlð1Þ¼ i 2ðH†Dμ ↔ HÞðlαγμlβÞ OαβHlð3Þ¼ i 2ðH†Daμ ↔ HÞðlαγμτal βÞ OαβeH¼ ðH†HÞlαHeβ Oαβγδll ¼ 12ðlαγμlβÞðlγγμlδÞ ð2:7Þ
2Note that this definition of the dimension-five operator is the
where we normalize the“Hermitian” operators with a factor of1=2 (see AppendixDfor a discussion) in order to agree with[19,20], and iðH†D μ ↔ HÞ ≡ iðH†DμHÞ − iðDμHÞ†H ¼ H†ði∂ μHÞ − ið∂μHÞ†H− gH†τaWaμH − 2YðHÞg0H†B μH; iðH†Da μ ↔ HÞ ≡ iðH†τaD μHÞ − iðDμHÞ†τaH: ð2:8Þ
The choice of operator basis implies a choice of operators that vanish by the equations of motion (EOMs). For example i=Dlα− ½YeασHeσ ¼ 0 implies that the follow-ing operators Oαβvð1Þ¼ iðH†HÞðl α=D ↔ lβÞ − ðH†HÞðl αHeσ½YTeσβþ ½YeασeσH†lβÞ; Oαβvð3Þ¼ iðH†τaHÞðl α=Da ↔ lβÞ − ðH†HÞðl αHeσ½YTeσβþ ½YeασeσH†lβÞ; ð2:9Þ
are EOM-vanishing operators. The role of these operators becomes clear by noting that in intermediate steps of our off-shell calculations, additional structures appear that can conveniently be matched onto combinations of EOM-vanishing operators and operators of the Warsaw basis. For example the structures involving two Higgs fields and a covariant derivative of a lepton doublet are expressed in terms of the above operators as:
SαβHDlð1Þ ¼ iðH†HÞðl α=D ↔ lβÞ ¼ Oαβvð1Þþ Oασ eH½YTeσβþ ½YeασO†σβeH; SαβHDlð3Þ ¼ iðH†τaHÞðl α=Da ↔ lβÞ ¼ Oαβvð3Þþ Oασ eH½YTeσβþ ½YeασO†σβeH: ð2:10Þ
In practice, if the coefficients CβαHDlð1Þ and CβαHDlð3Þ of these structures are present, they are equivalent to CβσeH¼ CβαHDlð1Þ½Yeασþ CβαHDlð3Þ½Yeασ (and the Hermitian
conjugate relation).
A. In the case of the 2HDM
In this section, we consider the addition of a second Higgs doublet H2 to the SM, of the same hypercharge as the SM Higgs (which we relabel H1). The LFV induced by double-insertions of dimension-five operators could be more significant in this model, because there are several dimension-five operators, so neutrino masses cannot con-strain them all. However, a complete analysis of LFV in the 2HDM would require extending the operator basis at dimension-six and calculating the additional terms in the RGEs, which is beyond the scope of this work. So for simplicity, we make three restrictions:
(1) First, we consider only the dimension-six LFV operators of the SMEFT. This is the appropriate set of dimension-six operators just above mW, provided that H2 has no vev, and that the mass m22 of the additional Higgses is sufficiently high: m2W ≪ m222 ≪ Λ2. In our phenomenological analysis we extend this range to the scenario m2W≲ m222≪ Λ2, by considering a Higgs potential where the additional Higgses are not directly observ-able at the LHC, and where the Yukawa couplings of H2 are vanishing. Such a scenario would e.g., be realized in the inert two Higgs doublet model[26– 29]and setting the scale m22close to the electroweak scale will not require the consideration of additional renormalization group effects in the SMEFT. (2) Second, we suppose that at the high scale Λ no
dimension-six LFV operators are generated. This is unrealistic, but allows us to focus on the LFV generated by double-insertions of the dimension-five operators. (3) Third, we suppose there is no LFV in the renorma-lizable couplings of the 2HDM (in particular, in the lepton Yukawas), so that when matching the 2HDM þ dimension-five operators onto the SMEFT at the intermediate scale m22, no additional LFV operators are generated.
Consider first the renormalizable Lagrangian. The Yukawa couplings can be written[30]:
δL2HDM¼ −ðν; eLÞ½Yð1Þ Hþ1 H01 e− e½Yð1Þ†H†1l − ðν; eLÞ½Yð2Þ Hþ2 H02 e− e½Yð2Þ†H†2l; ð2:11Þ
where the flavor indices are implicit, and the basis in (H1, H2) space is taken to be the “Higgs basis” where hH2i ¼ 0. We suppose that ½Yð1Þ and ½Yð2Þ are
simulta-neously diagonalizable on their lepton flavor indices. The second Yukawa coupling changes the equations of motion for the leptons, so the 2HDM version of the equation-of-motion vanishing operators [given in Eq. (2.9)for the single Higgs model] should be modified. As a result, the operatorsOHDlð1ÞandOHDlð3Þshould not be replaced only by the SMEFT operatorOeH, as given in Eq.(2.10), but also
by an operator with an external H2leg. However, since we neglect dimension-six operators with external H2, we use the relations(2.9)and(2.10)also in the 2HDM case.
In this“Higgs” basis, the most general Higgs potential is V¼ m211H†1H1þ m222H†2H2− ½m212H†1H2þ H:c: þ 1 2λ1ðH†1H1Þ2þ 12λ2ðH†2H2Þ2þ λ3ðH†1H1ÞðH†2H2Þ þ λ4ðH†1H2ÞðH†2H1Þ þ 1 2λ5ðH†1H2Þ2 þ ½λ6ðH†1H1Þ þ λ7ðH†2H2ÞH†1H2þ H:c: : ð2:12Þ
In order to decouple the additional Higgses, we can, for instance, set m212¼ 0 and assume m222≫ m2W, or leave m222 free, and impose m212¼ λ6¼ λ7¼ ½Yð2Þ ¼ 0.
At dimension-five in the 2HDM, there are four opera-tors [16]: δL ¼ þCαβ5 2ΛðlαεH1ÞðlcβεH1Þ þ Cαβ5 2Λ ðlcβεH1ÞðlαεH1Þ þCαβ21 2ΛððlαεH2ÞðlcβεH1Þ þ ðlβεH1ÞðlcαεH2ÞÞ þCαβ21 2Λ ððlcβεH2ÞðlαεH1Þ þ ðlcαεH1ÞðlβεH2ÞÞ þCαβ22 2ΛðlαεH2ÞðlcβεH2Þ þ Cαβ22 2Λ ðlcβεH2ÞðlαεH2Þ −CαβA 2ΛðlαεlcβÞðH†1εH2Þ − CαβA 2Λ ðlcβεlαÞðH2εH1Þ; ð2:13Þ where fC5; C22; C21g are symmetric on flavor indices (so can contribute to neutrino masses). In the O21 operator, ðlαεH2ÞðlcβεH1Þ ¼ ðlβεH1ÞðlcαεH2Þ, but both terms are
retained here because they are convenient in our Feynman rule conventions.3
Tree-level LFV is often avoided in the 2HDM by impos-ing a Z2 symmetry on the renormalizable Lagrangian: if under the Z2 transformation, H1→ H1 and H2→ −H2, then½Yð2Þ, λ6andλ7are forbidden. We will later discuss this case, but do not impose the Z2symmetry from the begin-ning, because it also forbids the C21and CAcoefficients at dimension-five.
III. THE EFT CALCULATION A. Diagrams and divergences
Diagrams with two insertions of the dimension-five operators are illustrated in Figs.1and2. We focus on the lepton flavor violating diagrams of Fig.1, and discuss the four-Higgs operators generated by Fig.2in Appendix G, because four-Higgs interactions are flavor conserving and arise in the SM.
The Feynman rules arising from the (tree-level) Lagrangian of equations (2.1), (2.4), (2.6) are given in AppendixA. We use them to evaluate, using dimensional regularization in4 − 2ϵ dimensions in MS, the coefficient of the 1=ϵ divergence of each diagram of Fig. 1. These coefficients can be expressed as a sum of numerical factors multiplying the Feynman rules for the dimension-six operators of Eqs. (2.7) and (2.10) (these Feynman rules are given in AppendixA), and then the EOMs are used to transform the operators of Eq.(2.10)toOeHandO†eH. The required counterterm ΔCO for each of the dimension-six
operators given in Eq.(2.7)can be identified asð−1Þ× the numerical factor that multiplies its Feynman rule. This counterterm is added in the Lagrangian to the operator coefficient CO, resulting in a “bare” coefficient CO;bare ¼ μ2ϵðC
Oþ ΔCOÞ that should be independent of the MS
renormalization scaleμ. Note that the factor μ2ϵ is chosen such that bare Lagrangian remains d-dimensional.
A more complete and rigorous presentation will be required in the next section, in order to derive the RGEs, so let us replace counterterms by Z factors in order to minimize notation and introduce the necessary factors of μ2ϵ to obtain the correct dimensions. More details of the
formalism and calculations are given in AppendixC. We allow for multiple operators at both dimension-six and -five, and align the dimension-six coefficients in a row vector ˜C, and the dimension-five coefficients in a row vector ⃗C. Then the bare coefficients can be written
FIG. 2. Two insertions of dimension-five operators can also contribute to dimension-six operators involving four Higgses via this diagram.
3The operatorO
21can also be written as2ðlβϵH1ÞðlcαϵH2Þ þ
ðlβϵlcαÞðH1ϵH2Þ using the identity (A9), as done in the first
⃗Cbare¼ μ2ϵ⃗C½Z; ˜Cbare¼ μ2ϵ½ ˜C ˆZ þ ⃗C½ ˜Z ⃗C† ð3:1Þ
where matrices wearing a hat act on the space of dimension-six coefficients, and matrices in square brackets act in the dimension-five space, so ˆZ represents the renormalization of dimension-six coefficients amongst themselves, and ½Z represents the renormalization of dimension-five coeffi-cients. The quantity ½ ˜Z renormalizes insertions of two dimension-five operators; ½ ˜Zijk is a vector in the dimen-sion-six space with index k, and a matrix in the dimension-five space with indices i, j. In the single Higgs model, i, j correspond to the flavor indices of the Weinberg operator, e.g., i¼ αβ, j ¼ ρσ. The index k corresponds to the operator labels and flavor indices of dimension-six operators. The counterterms that renormalize the diagrams of Fig.1are then components of the vector ⃗C½ ˜Z ⃗C†. All terms in the above expressions assume an implicit sum over flavor indices; the explicit flavor dependence is presented in AppendixB.
The first diagram of Fig. 1 has two Higgs and two doublet-lepton legs and so must be renormalized by the operatorsOHlð3ÞandOHlð1Þ, and the structuresSHDlð1Þand SHDlð3Þ. Since these all involve a derivative, the diagram is
calculated for finite external momenta. The counterterms that we obtain from this diagram differ from those given in
[21]; as discussed in AppendixF, it appears that the authors of [21] dropped one of the terms multiplying the 1=ϵ divergence. We check our result by attaching an external Bμ or Waμ boson, in all possible ways, to the first diagram of Fig. 1, and verify that our counterterms also cancel the divergences of the 2-Higgs-2-lepton-gauge boson vertices generated by two insertions of the Weinberg operator (this is outlined in Appendix B 3). This diagram can be renormalized using the following counterterms:
ð ⃗C½ ˜Z ⃗C†ÞβαHlð1Þ¼ −3 4 1 16π2ϵ½C5C5βα; ð3:2Þ ð ⃗C½ ˜Z ⃗C†ÞβαHlð3Þ ¼ þ 2 4 1 16π2ϵ½C5C5βα; ð3:3Þ ð ⃗C½ ˜Z ⃗C†ÞβαHDlð1Þ ¼ −3 4 1 16π2ϵ½C5C5βα; ð3:4Þ ð ⃗C½ ˜Z ⃗C†ÞβαHDlð3Þ¼ þ 2 4 1 16π2ϵ½C5C5βα; ð3:5Þ
where the last two counterterms represent divergences proportional to the structures SHDlð1Þ andSHDlð3Þ, which contribute to the renormalization of CeHthrough the linear
combination given in Eq. (2.10).
The middle diagram of Fig.1 contributes to OβαeH, and the divergence it induces can be removed by the counter-term ð16π2ϵÞ−1½C5C5Yβα (where the flavor index order is doublet-singlet). Including also the counterterms for SβαHDlð1Þ andSβαHDlð3Þ [Eqs. (3.4), (3.5)] gives
ð ⃗C½ ˜Z ⃗C†ÞβαeH ¼ þ 34
1
16π2ϵ½C5C5Yβα: ð3:6Þ
Since the structures SHDlð3Þ and SHDlð1Þ are Hermitian, they contribute to the renormalization of bothOeHandO†eH
[see Eq.(2.10)]. Only the contribution toOeHis included in
(3.6), because the Hermitian conjugate in(2.6)generates a counterterm proportional to O†eH that absorbs the diver-gence of the“conjugate” process of Fig. 1.
The third diagram of Fig.1contributes to the four-lepton operator Oραβσll , and the divergence it induces can be removed by the counterterm
ð ⃗C½ ˜Z ⃗C†Þραβσll ¼ −1 4
1
16π2ϵCρβ5 Cσα5 : ð3:7Þ
B. The 2HDM
In the 2HDM, we consider diagrams analogous to Fig.1, but with insertions of any of the dimension-five operators given in Eq.(2.13). The external Higgs lines are required to be H1, but the internal Higgs lines can be either doublet. The counterterms required to cancel double-insertions of the O5 operator, discussed in the previous section, also arise in the 2HDM. In this section, we only list the additional contributions to the counterterms.
We start again with the first diagram of Fig.1, withO21 orOA at the vertices. Since by construction, the Feynman rule forO21is identical to the rule forO5, double-insertions ofO21require the same counterterms as given in Eqs.(3.2)
to (3.5), but with C5, C5 replaced by C21, C21. Double insertions of the antisymmetric operator OA require the counterterms: Δð ⃗C½ ˜Z ⃗C†ÞβαHlð1Þ ¼ 1 4 1 16π2ϵ½CACAβα; ð3:8Þ Δð ⃗C½ ˜Z ⃗C†ÞβαHDlð1Þ ¼ 1 4 1 16π2ϵ½CACAβα: ð3:9Þ
Finally,OA at one vertex andO21 at the other require the
contributions to the counterterms: Δð ⃗C½ ˜Z ⃗C†ÞβαHlð3Þ¼ 1 4 1 16π2ϵ½CAC21− C21CAβα; ð3:10Þ Δð ⃗C½ ˜Z ⃗C†ÞβαHDlð3Þ¼ 1 4 1 16π2ϵ½CAC21− C21CAβα: ð3:11Þ
It is straightforward to check, using respectively the anti-symmetry and anti-symmetry of CAand C21 on flavor indices,
that the combination ½CAC21− C21CA is Hermitian, as
farthest from the Yukawa coupling, which can be cancelled by the countertermsð16π2ϵÞ−1½C21C5Yð2Þβαand −½CAC5Yð2Þβα=ð16π2ϵÞ. Including also the additional
counterterms forOβαHDlð1Þ andOβαHDlð3Þ in the 2HDM gives Δð ⃗C½ ˜Z ⃗C†ÞβαeH¼ 1 4 1 16π2ϵð4½ðC21− CAÞC5Yð2Þβα þ ½ðCACAþ CAC21− C21CA − C21C21ÞYð1ÞβαÞ: ð3:12Þ
Finally, for the four-lepton operator, there are additional counterterms in the 2HDM to cancel the divergences induced by double-insertions of O22, ofO21, and of OA. (The possible diagrams with an insertion of bothO21 and OA vanish due to antisymmetry.) We obtain:
Δð ⃗C½ ˜Z ⃗C†Þρσβαll ¼ −1 4 1 16π2ϵCρβ22Cασ22 −2116π12ϵCρβ21Cασ21 þ 1 2 1 16π2ϵCρβACασA : ð3:13Þ
C. The renormalization group equations The contribution of dimension-five operators to the renormalization group equations of dimension-six opera-tors, due to double insertions, can be obtained following the discussion of Herrlich and Nierste[31]. The derivation is presented in Appendix C. Here we schematically outline the result.
The bare Lagrangian coefficients are defined at one loop as in Eq.(3.1), where the counterterm for one operator can depend on the coefficients of other operators. Recall that the bare coefficients are independent of the dimensionful parameterμ, and that the renormalized Cs are dimension-less. Using ⃗C¼ μ−2ϵ⃗Cbare½Z−1 allows one to obtain, in
4 − 2ϵ dimensions: ð16π2Þμ d dμ⃗C ¼ −⃗C 2ϵð16π2Þ þ ð16π2Þ μ d dμZ ½Z−1 ≡ ⃗C½γ − 2ϵð16π2Þ ⃗C ð3:14Þ
where½γ denotes the 4-dimensional anomalous dimension matrix, and we (unconventionally)4 factor the16π2 out of the anomalous dimension matrices. While the −2ϵ term does not contribute in d¼ 4 dimensions to the mixing of the dimension-five operators, it plays an essential role in the renormalization group equations of the dimension-six operators.
For the dimension-six coefficients, it is straightforward to obtain from Eq.(3.1):
μ d dμ ˜C ¼ − ˜C · μ d dμ ˆZ ˆZ−1þ 2ϵ ⃗C · ˜Z · ⃗C†ˆZ−1 − ⃗C · μ d dμ˜Z · ⃗C†ˆZ−1 − ⃗C · ½Z μ d dμZ −1 ·½ ˜Z · ⃗C†ˆZ−1 − ⃗C · ½ ˜Z · μ d dμZ −1 † ½Z†⃗C†ˆZ−1; ð3:15Þ
where terms of OðϵÞ that vanish in 4 dimensions are neglected, and the summation over flavor and operator indices is indicated with a dot. The second line can be dropped, because the first term vanishes at one loop, and the remaining terms are of two-loop order because both½ ˜Z and d½Z−1=dμ arise at one-loop. So the renormalization group
equations for the dimension-six coefficients can be written ð16π2Þμ d
dμ˜C ¼ ˜C ˆγ þ⃗C½˜γ⃗C
†; ð3:16Þ
where ˆγ is the one-loop anomalous dimension matrix for dimension-six operators[19]and½˜γ ¼ 2ð16π2Þϵ½ ˜Z is the anomalous dimension tensor.
We give below the anomalous dimensions describing the one-loop mixing of double-insertions of dimension-five operators into LFV dimension-six operators, in the 2HDM. The single Higgs model can be easily retrieved by setting C21¼ CA¼ C22¼ 0 in the equations below. The
anoma-lous dimension tensor mixing a pair of dimension-five operators into a dimension-six operator is necessarily a three-index object; below we sum over the two dimension-five indices, and give these summed components of the tensor as elements of a vector in the dimension-six operator space. These anomalous dimensions parametrize the mix-ing of Fig.1 in the 2HDM (recall that a factor1=16π2 is scaled out of our anomalous dimensions):
ð ⃗C½˜γ ⃗C†ÞβαHlð1Þ¼ −Cβρ5 3δ2ρσCσα5 − Cβρ213δρσ 2 Cσα21 þ CβρA δρσ 2 CσαA ð3:17Þ ð ⃗C½˜γ ⃗C†ÞβαHlð3Þ¼ Cβρ5 δρσC5σαþ Cβρ21δρσCσα21 þ CβρA δρσ 2 Cσα21 − Cβρ21δρσ 2 CσαA ð3:18Þ ð ⃗C½˜γ ⃗C†ÞβαeH¼ Cβρ5 3½Yð1Þηαδρσ 2 Cση5 þ2½ðC21−CAÞC5Yð2Þβα þ1 2½ðCACAþCAC21−C21CA−C21C21ÞYð1Þβα ð3:19Þ 4
The usual definition[15]isμd
dμC¼ Cγ, then γ is expanded in
loops:γ ¼αs
4πγ0þ …. However, here we only work at one loop,
ð ⃗C½˜γ ⃗C†Þρσβαll ¼ −Cβρ5 1
2Cσα5 − Cβρ2212Cσα22
− Cβρ21Cσα21 þ CβρACσαA ð3:20Þ where the operator label and flavor indices on the left-hand side refer to the dimension-six operator (the dimension-five indices are summed).
In the next section, we will need the RGEs for dimension-five operators. Recall that in the single Higgs model, ½γ is in principle a 9 × 9 matrix (or 6 × 6, if one uses the symmetry of Cαβ5 ), mixing the elements of C5 among themselves. However, in the basis where the charged leptons are diagonal, ½γ is diagonal, and the anomalous dimension for the coefficient Cαβ5 of the Weinberg operator is [16]:
16π2γ ¼ −3
2ð½Ye2ααþ ½Yeββ2 Þ þ ðλ − 3g2þ 2Trð3½Yu†½Yu
þ 3½Yd†½Yd þ ½Ye†½YeÞÞ ð3:21Þ
where the Higgs self-interaction in the SM Lagrangian is
λ
4ðH†HÞ2, and ½Yf are the fermion Yukawa matrices.
IV. PHENOMENOLOGY
In order to solve the RGEs, it is convenient to define t¼16π12lnmμ
W, in which case the one-loop RGEs
for dimension-five and -six operator coefficients can be written as d dt˜C ¼ ˜C · ˆγ þ ⃗C · ½˜γ · ⃗C † d dt⃗C ¼ ⃗C · ½γ: ð4:1Þ
These are among the most familiar of differential equations, whose solutions have the form
⃗CðtfÞ ¼ ⃗Cð0Þexpfγtfg ≃ ⃗Cð0Þ 1 þ γ 1 16π2ln Λ mW þ … ð4:2Þ ˜CðtfÞ ¼ Z t f 0 dτ ⃗Cð0Þe γτ½˜γ½eγτT⃗C†ð0Þe−ˆγτþ ˜Cð0Þ eˆγtf ð4:3Þ where16π2tf¼ lnðmΛWÞ. In these solutions, the anomalous
dimension matrices were approximated as constant; this is not necessarily a good approximation, because the anoma-lous dimensions depend on running coupling constants, in particular the Yukawa couplings can evolve significantly above mW. Although the impact of the running Yukawa couplings is a higher order effect if the logarithms are not
resummed, their contribution could be amplified by a large logarithm. Therefore, if one wished to perform a precision prediction of lepton flavor violation in the case thatΛ ⋙ v, it would be necessary to incorporate the effect from running Yukawa couplings. However, this mass hierarchy highly suppresses lepton flavor violation, and such a precision prediction is beyond the scope of this work.
A simple solution to Eq. (4.3) can be obtained by expanding the exponentials under the integral, as in Eq.(4.2): ˜CðmWÞ ¼ ˜CðΛÞ − ˜CðΛÞˆγ 1 16π2ln Λ mW − ⃗CðΛÞ½˜γ ⃗C†ðΛÞ 1 16π2ln Λ mW þ ð4:4Þ
A. The single Higgs model
In the SM case where there is only one Higgs doublet, there is only the Weinberg operator at dimension-five: a symmetric3 × 3 matrix, whose entries are determined by neutrino masses and mixing angles (in the mass basis of charged leptons). We now want to estimate the contribution of double-insertions of this dimension-five operator to lepton-flavor violating processes.
We neglect the “Majorana phases,” suppose that the lightest neutrino mass is negligible, and neglect the lepton Yukawas in the RGEs. Then the RG running of Cαβ5 between mW and Λ can be approximated as a rescaling,
withγ ≈ λ − 3g2þ 6y2t ≈ 3.5: Cαβ5 ðΛÞ ¼ Cαβ5 ðmWÞ 1 þ 3.5 1 16π2ln Λ mW þ ð4:5Þ For Λ ≤ 1016 GeV, the log is≤ 32.
We can now estimate the contribution of the neutrino mass operator to lepton flavor violating processes from Eq.(4.4). We neglect ˜CðΛÞ and find that the contribution is
1 16π2lnmΛ
W× the coefficients of Eqs.(3.17)–(3.20), that is, of
order ˜CðmWÞ ∼ C25 16π2lnmΛ W : ð4:6Þ
As expected, this is negligibly small, because C25=Λ2∼ m2ν=v4. Notice, however, that the logarithm can be large, and the anomalous dimension is Oð1Þ; the effect is tiny because small neutrino masses imply that C5is small orΛ is large.
B. The two Higgs doublet model
suppressed by the smallness of the neutrino masses. The situation changes in an extended Higgs sector, where more than one dimension-five operator is present. The operator OA cannot contribute to neutrino masses as it is
antisym-metric in flavor space and is hence unconstrained. In addition, the neutrino mass contribution of operatorsO21 andO22 is suppressed if the vacuum expectation value of the second Higgs doublet is small. Renormalization group effects [16–18] will in general mix all operators, which could lift these suppression mechanisms at loop level. However the mixing factorizes in the limit whereλ6,λ7and Yð2Þ tend to zero: then the operatorsO21 andOA will not
mix intoO5andO22and are hence not constrained by the observed neutrino masses. Furthermore, the mixing ofO22 intoO5vanishes in the limit where in additionλ5tends to zero (see [32]for a symmetry argument).
In the following we will study the sensitivity of lepton-flavor violating decays to these additional operators. We assume that the Wilson coefficients of the dimension-five operators are generated at Λ ¼ 10 TeV, while all other dimension-six Wilson coefficients are zero at this scale. To avoid constraints from the observed neutrino masses we consider the scenario where the second Higgs doublet has a negligible vacuum expectation value and a mass at the weak scale. The Higgs sector could be assumed to be close to that of an inert two-Higgs doublet model[26–29]and the dangerous couplings λ6, λ7 and Yð2Þ are not generated radiatively. Renormalization group running will then gen-erate nonzero Wilson coefficients of several dimension-six operators atμ ∼ v. Only those dimension-six operators that involve standard model particles are of interest to us, since the vanishing vacuum expectation value of the second Higgs doublet will suppress the contribution of the other operators after spontaneous symmetry breaking. Applying the con-straints presented in AppendixEof the Wilson coefficients evaluated using Eq. (4.4) neglecting the small log lnðm22=mWÞ, we find the following: the μ → 3e decays
provide the greatest sensitivity to the additional dimension-five Wilson coefficients. In particular the left-handed con-tribution implies Cee 21Ceμ21 þ 0.5Cee22Ceμ22 þ 0.1 X σ ðCeσ A − Ceσ21ÞðC σμ A þ C σμ 21 Þ < 1 5.2lnðΛ=m22Þ Λ 10 TeV 2 ; ð4:7Þ
where we neglected the mixing of the dimension-five operators amongst themselves, as this would contribute at two-loop order to the lepton flavor violating processes. For the right-handed contribution we find
X σ ðCeσ A − Ceσ21ÞðC σμ A þ C σμ 21 Þ <lnðΛ=m1.6 22Þ10 TeVΛ 2; ð4:8Þ
which exhibits a weaker sensitivity. The contribution of the μ ↔ e flavor-changing Z vertex to μ → eγ is relatively suppressed by a loop factor, so is beyond current exper-imental sensitivity. However, this Z vertex contributes at tree-level to μ → e conversion, in interference with vector and scalar 2-quark-2-lepton operators. Indeed, the current sensitivity of μ → e conversion in gold is jCeμHlð1Þþ CeμHlð3Þj ≈ 1.4 × 10−7ðΛ=m
tÞ2. The resulting
con-straints on the Wilson coefficients reads: X σ ðCeσ A −Ce21σÞðC σμ A þC σμ 21 Þ <6.5lnðΛ=m1 22Þ10TeVΛ 2; ð4:9Þ We also checked that the current experimental situation forτ decays does not lead to significant constraints.
V. SUMMARY
Motivated by neutrino masses and the expected progress in searches for lepton flavor violation, we calculated the leading one-loop contribution of a pair of lepton number violating dimension-five operators to the coefficients of lepton flavor violating dimension-six operators. The diagrams are given in Fig. 1. The dimension-five operators that we considered are the Weinberg operator, constructed out of SM fields and given in Eq. (2.4), and three additional dimension-five operators that can be constructed in the two Higgs doublet model, given in Eq. (2.13). The dimension-six, lepton flavor violating operators of the SMEFT are listed in Appendix D, in the “Warsaw” basis, and the subset of these operators relevant for our calculation is given in Eq. (2.7). A selection of constraints on their coefficients, evaluated at the weak scale, is given in Appendix E.
model couplings, so, in an expansion in terms of SM couplings, our result is the “leading” contribution to the one-loop RGEs of the dimension-six SMEFT.5
In the effective field theory constructed with standard model fields, the coefficient of the Weinberg operator is proportional to the neutrino mass matrix. So the lepton flavor changing amplitudes induced by double insertions of the Weinberg operator are ∝ ðmν=mWÞ2lnΛ=mW, and far below current sensitivities. This is outlined in Sec. IVA. However, the situation is different in the 2 Higgs doublet model, as discussed in Sec.IV B: there are four operators at dimension five, and the neutrino mass matrix only constrains one combination. We evaluated the mixing of the four operators into lepton flavor violating operators of the standard model effective theory, and for a lepton number violating scale of 10 TeV we found that the current experimental value of μ → 3e is sensitive to the Wilson coefficients of these additional operators.
ACKNOWLEDGMENTS
The research of M. G. was supported in part by U.K. Science and Technology Facilities Council (STFC) through the Consolidated Grant No. ST/L000431/1. M. G. thanks the MITP workshop on “Low-Energy Probes of New Physics” for hospitality and a stimulating work environ-ment. M. L. would like to thank the Les Houches Summer School Session CVIII for much useful and interesting discussion, and a wonderful environment in which part of this work was done. We thank Rupert Coy for drawing our attention to Ref. [21].
APPENDIX A: FEYNMAN RULES AND IDENTITIES
1. Feynman rules
We use Feynman rules of Ref. [25], in order that the fermion traces in loops multiply spinors in the correct order. The Feynman rule for the Weinberg operator of Eq. (2.4)
can be obtained reliably by using Lehmann-Symanzik-Zimmermann reduction or Wick’s theorem, which gives the signs for fermion interchange. The fermion fields are expanded as [33] ψðxÞ ¼X s Z d3k ð2πÞ3 1ffiffiffiffiffiffi 2E p ðas
kusðkÞe−ik·xþ bs†k vsðkÞeþik·xÞ
so the amplitude Mfiis
hlαjHIji
Cσρ5
2ΛðlσnεnNHNÞðlcρmεmMHMÞjlcβiHJi
¼ ð−iÞiCαβ5
2ΛðuαjPRuβiþ uβiPRuαjÞðεiIεjJþ εiJεjIÞ
¼ ð−iÞiCαβ5 þ Cβα5
2Λ uαjPRuβiðεiIεjJþ εiJεjIÞ
¼ ð−iÞiCαβ5
Λ uαjPRuβiðεiIεjJþ εiJεjIÞ; ðA1Þ
where the SU(2) lepton indices are lower case, Higgs indices are upper case,lαjand lc
αjrepresent a final state
lepton and an initial state anti-lepton respectively. The factor i is the usual factor for Feynman rules and the factor ð−iÞ is due to the calculation of Mfi. This expression
agrees with Feynman rule of Ref.[21].
A Feynman-rule to attach a W-boson to thelc line also
will be needed. With the following identities[25]
lc¼ ClT; C¼ iγ 0γ2; C−1¼ C†; C†γμTC¼ −γμ ðA2Þ lc¼ ½CγT 0l†γ0¼ lTγ0C†γ0¼ lTC†Cγ0C†γ0 ¼ −lTC†γ 0γ0¼ −lTC−1 ðA3Þ
one obtains (where the (−1) is for interchanging fermions) ½liτijW=PLljT ¼ ð−1Þ½−lcjCτjiaPTLWaμγTμC−1lc ðA4Þ
¼ lcτaWa
μCγTμC−1PRlc ðA5Þ
¼ −lcτaWa
μγμPRlc ðA6Þ
and recall thatτ ¼ τ†, soτ¼ τT. The Feynman rules for the
four, five and physical dimension-six interactions are given in Figs.3,4, and5respectively. The Feynman rules for the dimension-six equation of motion vanishing interactions are given in Fig.6.
2. Identities The following identities are useful:
2εiIεjJ¼ δijδIJ− τaijτa;IJ Fierz ðA7Þ
1 4τaijτa;kl¼ 12δilδkj− 1 4δijδkl SUðNÞ ðA8Þ εabεcdþ εbcεadþ εacεbd¼ 0 ðA9Þ εiJεkJ ¼ δik ðA10Þ εijSajkεkl¼ Sali ðA11Þ 0 ¼ δijSakl− δjlSakiþ δklSaji− δikSajl ðA12Þ εijεkl¼ δikδjl− δilδjk; ðA13Þ where
5Mixing among dimension six operators occurs via the
FIG. 3. Feynman rules for dimension-four interactions.
ε ¼ 0 1 −1 0 ; ⃗τ ¼0 1 1 0 ;0 −i i 0 ;1 0 0 −1
and the SU(2) generators are Sa¼ τa=2.
APPENDIX B: THE LOOP CALCULATION 1. Flavor dependence
We allow for multiple operators at both dimension-five and -six, and denote a particular Wilson coefficient by CζX, where X andζ are the operator and flavor labels, respectively. Then the bare Wilson coefficients of the dimension-six standard model effective theory Lagrangian can be written as X ζ;X CζX;bareQζX;bare ¼ μ2ϵX θ;Y X ζ;X CζXZζθXYþX ζ;η Cζ5½Cη5†Zζηθ 55;Y QθY;bare; ðB1Þ
whereζ, η and θ represent generation indices of an operator, and the renormalization constants ZζθXYencode the mixing of dimension-six Wilson coefficients amongst themselves, which can be extracted from the anomalous dimensions of reference
[19]. In the standard model, the mixing of two dimension-five Wilson coefficients into a dimension-six coefficient is given by Zζηθ
55;Y. They are induced by the double-insertions of
dimen-sion-five operators, as shown in Fig.1. In the case of a 2HDM effective field theory we extend the summation of the five flavor indices to a sum over all dimension-five operators and their respective flavor components.
The renormalization constants can be expanded in the number of loops and powers of epsilon. At one-loop in the MS scheme the counterterms of the physical and EOM-vanishing operators are pure1=ϵ poles, and the renormal-ization of evanescent operators does not play a role. Hence we can expand
Zζηθ
55;j¼ 116π2
1
and write the generation summation in the case of an operator involving four fermions explicitly as:
Cζ5Cη†5 δZζηθ
55;XQθX ¼ Cαβ5 C5δγδZαβγδ;ρστυ55;X QρστυX : ðB3Þ
The sum over generation indices reduces trivially for operators that involve less fermions. The corresponding renormalization equation ensures that the pole of the one-loop off-shell matrix element of an insertion of two dimension-five operators is cancelled by its counterterm. Factoring out the common overall factor Cαβ5 Cδγ5 we write: hfjQαβ5 ðQγδ5Þ†jiijð1Þ
1=ϵþ ðδZαβγδ;ρστυ55;X hfjQρστυX jii þ H:c:Þ ¼ 0;
ðB4Þ wherejð1Þ1=ϵdenotes the1=ϵ pole of a one-loop diagram and hfj and jii are arbitrary off-shell final and initial states.
In calculations of the loop diagrams the following generation structures arose:
Tαβγδ;ρσ1 ¼ 1 2ðδδσδαρδβγþ δαδδβρδγσÞ; Tαβγδ;ρσ1A ¼ 1 2ðδδσδαρδβγ− δαδδβρδγσÞ; Tαβγδ;ρσ2 ¼ 1 2ðδαρδβγYδσ þ δαδδβρYγσÞ; Tαβγδ;ρσ2A ¼ 1 2ðδαρδβγY ð2Þ δσ − δαδδβρYð2ÞγσÞ; Tαβγδ;ρστυ3 ¼ 1 4ðδδσδγυþ δδυδγσÞðδατδβρþ δαρδβτÞ; Tαβγδ;ρστυ3A ¼ −1 4ðδδσδγυ− δδυδγσÞðδατδβρ− δαρδβτÞ: ðB5Þ These structures, when contracted with Wilson coefficients and multiplied by loop factors and the appropriateδZs, give the results in Eqs.(3.2)–(3.13). Note that in intermediate steps, more general structures than the T1ðAÞ, T2ðAÞlisted here
arise, which show a greater degree of symmetry. These can be obtained by taking the T1ðAÞ and T2ðAÞ given here, and symmetrizing/antisymmetrizing over the appropriate indi-ces, such that the structures become symmetric (or antisym-metric) under α ↔ β and symmetric (or antisymmetric) underγ ↔ δ. The structures in Eq.(B5)were matched onto the generation structures of the dimension-six operators (the matching is more subtle for the four-lepton operatorOαβγδll , where the matching is done via a Fierz-evanescent dimen-sion-six operatorOαβγδe ), and the generation structure
there-fore extracted from the renormalization constants, which can then be written as a generation structure multiplied by a numerical factor.
At one-loop we find the following nonvanishing mixing into the physical dimension-six operators
δZαβγδ;ρσ55;Hlð1Þ¼ −3 4Tαβγδ;ρσ1 ; δZαβγδ;ρσ2121;Hlð1Þ¼ −34Tαβγδ;ρσ1 ; δZαβγδ;ρσAA;Hlð1Þ¼ −1 4Tαβγδ;ρσ1 ; δZαβγδ;ρσ55;Hlð3Þ¼ 12Tαβγδ;ρσ1 ; δZαβγδ;ρσ2121;Hlð3Þ¼ 1 2Tαβγδ;ρσ1 ; δZαβγδ;ρσA21;Hlð3Þ¼ 14Tαβγδ;ρσ1A ; δZαβγδ;ρσ21A;Hlð3Þ¼ 1 4Tαβγδ;ρσ1A ; δZαβγδ;ρσ55;eH ¼ 34Tαβγδ;ρσ2 ; δZαβγδ;ρσ215;eH ¼ Tαβγδ;ρσ2 ; δZαβγδ;ρσ A5;eH ¼ −T αβγδ;ρσ 2A ; δZαβγδ;ρσAA;eH ¼ −14Tαβγδ;ρσ2 ; δZαβγδ;ρσ A21;eH ¼ 14T αβγδ;ρσ 2A ; δZαβγδ;ρσ21A;eH ¼ 1 4Tαβγδ;ρσ2A ; δZαβγδ;ρσ2121;eH¼ −14Tαβγδ;ρσ2 ; δZαβγδ;ρστυ55;ll ¼ −1 4Tαβγδ;ρστυ3 ; δZαβγδ;ρστυ2222;ll ¼ −14Tαβγδ;ρστυ3 ; δZαβγδ;ρστυ2121;ll ¼ −12Tαβγδ;ρστυ3 ; δZαβγδ;ρστυ AA;ll ¼ 12T αβγδ;ρστυ 3A : ðB6Þ
2. Four-lepton Green’s function
In the following we will explicitly present the renormalization of a Green’s function involving four lepton doublets. When we consider double-insertions of dimension-five operators one additional operator that vanishes in the limit d→ 4, a so-called evanescent operator, appears in our calculation. The exact defini-tion of the evanescent operator in d dimensions is not important, but will induce a scheme dependence beyond one-loop. We use
Oαβγδeva ¼ 1
2δijδklðliαlckγÞðlclδljβÞ −
1
2Oαβγδll ; ðB7Þ
where the first term has a left-right chirality structure and i, j, k, l are SU(2) indices.
Denoting the flavor and SU(2) component of the final statehfj ¼ hlk;ϕll;χj and the initial state jii ¼ jli;ψlj;ωi by ϕ, χ, ψ, ω, and i, j, k, l respectively, we find for the third diagram of Fig.1
hfjOαβ5 ðOγδ5Þ†jiijð1Þ 1=ϵ
¼ðuψiPLvωjÞðvϕkPRuχlÞ
64π2 ðδψδδωγþ δωδδψγÞ
×ðδχαδϕβþ δϕαδχβÞðδilδjkþ δikδjlÞ; ðB8Þ
which exactly matches the scalar contribution of the evanescent operatorOeva at tree level
hfjðδZαβγδ;ρστυ55;eva Oρστυeva;scalarþ H:c:ÞjiiLR¼ δZαβγδ;ρστυ55;eva ðuψiPLvωjÞðvϕkPRuχlÞ
×½δilδjkðδψσδωυδχρδϕτþ δωσδψυδϕρδχτÞ þ δikδjlðδωσδψυδχρδϕτþ δψσδωυδϕρδχτÞ;
ðB9Þ where we have used the hermiticity condition of the
renormalization constants.6 The one-loop contribution to the L × R part is then renormalized by the renormalization constantδZαβγδ;ρστυ
55;eva ¼ −12Tαβγδ;ρστυ3 . As there is noðV − AÞ ×
ðV − AÞ contribution to the Green’s function, the ðV − AÞ × ðV − AÞ parts have to cancel between the counterterms
of Oeva and Oll, i.e., δZαβγδ;ρστυ55;ll ¼ ð1=2ÞδZαβγδ;ρστυ55;eva ¼
−1
4Tαβγδ;ρστυ3 .
3.W emission
The values of renormalization constants may be checked by renormalizing other loop processes involving a double-insertion of dimension-five operators, and matching them to the same operator basisOβαHlð1Þ,OβαHlð3Þ,Oβαvð1ÞandOβαvð3Þ. The internal Higgs and lepton lines of the loop diagram
6The four-lepton renormalization constants fulfil the
may couple to Bμor Waμbosons of the Uð1ÞYand SUð2ÞL groups respectively. Since the group structure of Uð1ÞYis trivial, we concentrate here on the calculation resulting from emission of a Wa
μ boson. The results for emission of
Bμ emission may be retrieved from these results by replacing the SUð2ÞL generators everywhere by Uð1ÞY generators, 12τaij→ YðH; lÞδij at the beginning of the
calculation.
The renormalization equation for the process HMln α→ HJli βWaμ in MS is 0 ¼ hli βHJWaμjOγδ5ðO5ηκÞ†jlnαHMijð1Þ1 ϵ þ Zγδηκ;βα55;Hlð1Þð−g2Þ½uβiγμPLuαnτaJMδin þ Zγδηκ;βα 55;Hlð3Þð−g2Þ½uβiγμPLuαnðδJMτainÞ þ Zγδηκ;βα55;vð1Þð−g2Þ½uβiγμPLuαnδJMτain þ Zγδηκ;βα55;vð3Þð−g2Þ½uβiγμPLuαnðδinτaJMÞ:
where the tree-level matrix elements are replaced by their respective amplitudes and the SU(2) algebra has been simplified.
Two diagrams must be evaluated for the double insertion of dimension-five operators with associated emission of a Wa
μboson, which can couple to either the internal Higgs or
internal lepton. These diagrams are denoted byD1andD2, and are shown in Fig. 7.
Calculating the diagrams and isolating the 1=ϵ poles gives D1j1 ϵ ¼ 1ϵ g2 64π2½uβiγμPLuαnðδακδβγδδηÞ ×ð2δJMτain− δJnτaiM− δiMτaJn− δinτaJMÞ; ðB10Þ D2j1 ϵ ¼ 1ϵ g2 64π2½uβiγμPLuαnðδακδβγδδηÞ ×ðδiMτaJnþ δJnτaiM− 3δJMτainÞ; ðB11Þ
where we use the symmetries γ ↔ δ and η ↔ κ of the flavor indices of the Weinberg operator to simplify our
expressions here and in the following. The total amplitude of the double-insertion of dimension-five operators is therefore: hli βHJWaμjOγδ5ðO5ηκÞ†jlnαHMijð1Þ1 ϵ ¼ −1 ϵ g2 64π2½uβiγμPLuαnðδακδβγδδηÞðδJMτainþ δinτaJMÞ: ðB12Þ In this form it is simple to set up simultaneous equations for the renormalization condition by comparing the loop and tree amplitudes, Zγδηκ;βα 55;Hlð1Þð−g2Þ þ Z γδηκ;βα 55;vð3Þð−g2Þ − 1 ϵ g2 64π2ðδακδβγδδηÞ ¼ 0; ðB13Þ Zγδηκ;βα 55;vð1Þð−g2Þ þ Z γδηκ;βα 55;Hlð3Þð−g2Þ − 1 ϵ g2 64π2ðδακδβγδδηÞ ¼ 0: ðB14Þ This underconstrained set of equations may be constrained by substituting in solutions for Zγδηκ;βα
55;vð1Þand Z γδηκ;βα
55;vð3Þfrom the
momentum-dependent calculation, to verify the solutions δZγδηκ;βα55;Hlð1Þ¼ −3
4Tκβγδ;αη1 ; δZγδηκ;βα55;Hlð3Þ¼ 12Tκβγδ;αη1 : ðB15Þ
APPENDIX C: RENORMALIZATION GROUP EQUATIONS
The bare Wilson coefficients of dimension-five operators can be written as
⃗Cη
X;bare ¼ μ2ϵ⃗C θ
YðμÞZθηYXðμÞ; ðC1Þ
where ⃗CθYðμÞ is the renormalized Wilson coefficient,
ZθηYXðμÞ is the renormalization matrix, and μ is the renorm-alization scale. The μ2ϵ introduces an additional term proportional to ϵ into the d-dimensional renormalization group equation μ d dμ⃗C η X ¼ − ⃗CθY μ d dμZ θζ YZ ½Z−1ζη ZX− 2ϵ ⃗CηX: ðC2Þ
This reduces to the renormalization group equation in d¼ 4 dimensions ð16π2Þμ d dμ⃗C η X ¼ d¼4⃗Cθ YγθηYX; ðC3Þ
where the 4-dimensional anomalous dimension matrix
FIG. 7. Double insertions of dimension-five operators with associated emission of Wa
γθηYX ¼ −ð16π2Þ μ d dμZ θζ YZ ½Z−1ζη ZX ðC4Þ
is independent of the choice of the overall factor μ2ϵ. Therefore theμ2ϵ term can be neglected when only consid-ering mixing amongst operators of equal dimensions. In the case of mixing between operators of different dimensions a more careful treatment is required.
At loop level, operators of different dimensions can mix via multiple operator insertions[31]. Consider the specific case of loop diagrams involving two dimension-five oper-ators mixing into diagrams with a single dimension-six operator insertion. We denote dimension-six quantities with a tilde, quantities that mix dimension-five and -six with a hat, and dimension-five quantities without a tilde or hat. The bare dimension-six Wilson coefficient is
˜Cη
X;bare¼ μ2ϵ˜CθYðμÞ ˆZθηYXðμÞ þ μ2ϵCζAðμÞ ˜Z ζθ;η
AB;XðμÞ½CθB†ðμÞ;
ðC5Þ where ˜Cbare is μ-independent. Therefore the
renormaliza-tion group equarenormaliza-tion is ð16π2Þμ d dμ ˜C η X¼ ˜CθYˆγ θη YXþ C ζ A˜γ ζθ;η AB;X½CθB†; ðC6Þ
where ˆγθηYX is defined analogously to Eq. (C2), and ˜γζθ;ηAB;X¼ ð16π2Þ 2ϵ ˜Zζθ;υAB;Y−μ d dμ˜Z ζθ;υ AB;Y ½ ˆZ−1υη YX − ð16π2Þð½γθω BD†δ ζχ ACþ γ ζχ ACδθωBDÞ ˜Z χω;υ CD;Y½ ˆZ−1 υη YX ðC7Þ
where the explicit form in terms of generation indices is ½γαβγδAB †¼ ½γ
βαδγ
AB andδ αβγδ
AB ¼ δABδαγδβδ. The terms in the
second line of the above equation only contribute beyond one-loop. Furthermore, the contribution to the renormali-zation tensor Zζθ;υAB;Y isμ independent at one-loop and only the term proportional to 2ϵ contributes in our calculation. A comment regarding the sign of the2ϵ contribution is in order. The factor in μ2ϵ in (C5)generates a term propor-tional to −2ϵ, while the derivative of the dimension-five Wilson coefficients generates a contribution proportional to 2 × 2ϵ from (C2). Hence the one-loop anomalous dimen-sion matrix reads
˜γζη;θAB;C¼ 2δ ˜Z ζη;θ
AB;C ðC8Þ
in terms of the one-loop renormalization constants defined in Eq.(B2). Correspondingly we find½˜γ ¼ 2ð16π2Þϵ½ ˜Z.
APPENDIX D: OPERATORS
This Appendix lists dimension-six, SM-gauge invariant operators that change lepton flavor.The operators are in the
Buchmuller-Wyler basis, as pruned in Grzadkowski et al.
[20], commonly referred to as the “Warsaw” basis. All operators are added to the Lagrangianþ H:c:, as given in Eq.(2.6): δL6¼ X X;ζ CζX Λ2OζXþ H:c:
where the flavor indices are represented byζ, and are all summed over all generations. In the conventions of [20]
and[19], the hermitian conjugate is not added for “self-conjugate” operators, for whichPζCζXOζX¼ ½PζCζXOζX†. (For instance, Oαβρσll of Eq. (D11) is Hermitian, because ½ðeγμμÞðτγ
μτÞ†¼ ðμγμeÞðτγμτÞ). So we define such
oper-ators with a factor1=2 to avoid this double-counting. The four-fermion operators involving β ↔ α flavor change and two quarks are
Oð1Þαβnmlq ¼ 1 2ðlαγμlβÞðqnγμqmÞ ðD1Þ Oð3Þαβnmlq ¼ 1 2ðlαγμτalβÞðqnγμτaqmÞ ðD2Þ Oαβnmeq ¼ 1 2ðeαγμeβÞðqnγμqmÞ ðD3Þ Oαβnmlu ¼ 1 2ðlαγμlβÞðunγμumÞ ðD4Þ Oαβnmld ¼ 1 2ðlαγμlβÞðdnγμdmÞ ðD5Þ Oαβnmeu ¼ 1 2ðeαγμeβÞðunγμumÞ ðD6Þ Oαβnmed ¼ 12ðeαγμeβÞðdnγμdmÞ ðD7Þ Oαβnmlequ ¼ ðlA αeβÞεABðqBnumÞ ðD8Þ Oαβnmledq ¼ ðlαeβÞðdnqmÞ ðD9Þ OαβnmT;lequ¼ ðlAασβνeβÞεABðqBnσβνumÞ ðD10Þ
where l, q are doublets and e, u are singlets, n, m are possibly equal quark family indices, and A, B are SU(2) indices. The operator names are as in[20]withφ → H; the flavor indices are in superscript.
Oαβρσll ¼ 1 2ðlαγμlβÞðlργμlσÞ ðD11Þ Oαβρσle ¼ 1 2ðlαγμlβÞðeργμeσÞ ðD12Þ Oαβρσee ¼ 1 2ðeαγμeβÞðeργμeσÞ: ðD13Þ Then there are the operators allowing interactions with gauge bosons and Higgses. This includes the dipoles, which are normalized with the muon Yukawa coupling so as to match onto the normalization of Kuno-Okada[2]: OαβeH¼ ðH†HÞðlαHeβÞ ðD14Þ OαβeW¼ yβðlατaHσμνeβÞWaμν ðD15Þ OαβeB¼ yβðlαHσμνeβÞBμν ðD16Þ OαβHlð1Þ¼ i 2ðH†Dμ ↔ HÞðlαγμlβÞ ðD17Þ OαβHlð3Þ¼ i 2ðH†Daμ ↔ HÞðlαγμτal βÞ ðD18Þ OαβHe¼ i 2ðH†Dμ ↔ HÞðeαγμe βÞ; ðD19Þ
where yβdenotes the Yukawa coupling of a charged lepton eβin the mass basis, the double derivatives are defined in Eq. (2.8), and we include factors of 1=2 for Hermitian operators as discussed above Eq.(D1).
APPENDIX E: EXPERIMENTAL BOUNDS ON COEFFICIENTS
The aim of this Appendix is to obtain experimental constraints on the coefficients of the LFV operators of Eq.(2.7), evaluated at the weak scale mW. We are interested in this subset of operators because they are generated at one loop by double-insertions of dimension-five, lepton number changing (LNV) operators. Such constraints will allow an estimation of the sensitivity of LFV processes to the coefficients of LNV operators. We neglect the constraints on 2-lepton-2-quark operators, which are beyond the scope of this work, and focus onτ ↔ e and τ ↔ μ flavor change, becauseμ ↔ e is discussed in[34,35]. Nonetheless, some μ ↔ e bounds are listed for completeness.
Recall that constraints and sensitivities are different. A constraint is an exclusion, which tells the range of values a coefficient cannot have. For instance, the dipole coefficient (evaluated at the muon mass scale) CeμD;RðmμÞ, cannot be larger than 1.05 × 10−8 because the branching ratio searched for by the MEG experiment [3]is
BRðμ → eγÞ ¼ 384π2v
4
Λ4ðjCeμD;Lj2þ jC eμ D;Rj2Þ;
and the current experimental search imposes this constraint. Sensitivity is often discussed when an observable depends on many coefficients, and gives the range of values where a coefficient could have been seen. For instance, among the many loop processes that contribute toμ → eγ, there are two-loop diagrams involving flavor-changing Higgs cou-pling CeμeHðmWÞ. Calculating these diagrams and imposing
that they saturate the current experimental bound gives eαeyt 8π3y μ CeμeHðmWÞ ¼ 1.05 × 10−8;
where the Yukawa eigenvalue of fermion f is denoted yf.
Smaller values of CeμeH are allowed (the experiment could not have seen them), but larger values are not excluded by MEG, because many other operator coefficients could contribute to the rate, with possibly cancellations.
The difference between an exclusion and a sensitivity is illustrated in Fig.8, where the allowed region is the diagonal ellipse. The horizontal variable x is excluded outside the projection of the ellipse onto the x-axis (where the axis is thickened). But the experiment is only insensitive to x inside the intersection of the axis with the ellipse (dashed red line). Values of x between these two regions are allowed, provided that y has the appropriately correlated value.
Three ways to relate low-energy experimental bounds to the coefficients of operators at a higher scale are:
(1) to calculate the sensitivity of an experimental proc-ess to a particular operator coefficient. This is usually simple.
(2) To express an experimental rate as a function of high-scale coefficients. This is slightly more diffi-cult, because more coefficients are involved: each coefficient that contributes at the experimental scale will become a linear combination of high scale coefficients due the renormalization group mixing.
x -1.5 -1 -0.5 0 0.5 1 1.5 y -1 -0.5 0 0.5 1
(3) To obtain constraints on coefficients at the high scale. This is more involved, because a sufficient number of experimental constraints must be com-bined, in order to obtain a finite allowed region in coefficient space (no “flat directions”). Then the allowed region must be projected onto the various axes, in order to obtain constraints.
The third option is the most useful, but beyond the scope of this work. Instead here, we partially follow the second option, as a contribution to the third: we consider experimental bounds on the dimension-six operators which are generated in RGE evolution by double-inser-tions of dimension-five operators that change lepton
number. We aim to quote these bounds at mW. The
processes in question are LFV Higgs and Z decays (which occur at the weak scale), and flavor-changing lepton decays at low energy (these bounds must be translated to the weak scale via the RGEs of QED and QCD). So we will not succeed in our aim of setting constraints on coefficients at mW, because the low-energy experimental bounds depend on many coefficients at the weak scale, and we do not include enough experimental bounds.
In the following sections, we outline the calculations of the various rates, and summarize the experimental con-straints on coefficients at mW in Table I.
TABLE I. Bound on operator coefficients of the SMEFT, evaluated at mW, from the bounds listed in column 2 on the processes of
column 1. The bounds on coefficients of Hermitian operators (OHlð1Þ,OHlð3Þ,Oll,Ole) also apply to the conjugate coefficient. All the
bounds apply to running coefficients evaluated at mW, and are forΛ ¼ v ≃ mt. The combination of coefficients Cpenguinis defined in
Eq.(E12)and before Eq.(E23),δ is defined after Eq.(E23), and ge
R¼ 2s2W, geL¼ −1 þ 2s2W. Process BR < v2 Λ2j PCj < Z→ eμ∓ 7.5 × 10−7[4] jCeμHlð1Þþ CeμHlð3Þj < 1.2 × 10−3 Z→ τμ∓ 1.2 × 10−5[5] jCμτHlð1Þþ CμτHlð3Þj < 4.6 × 10−3 Z→ eτ∓ 9.8 × 10−6[6] jCeτ Hlð1Þþ CeHlð3Þτ j < 4.1 × 10−3 h→ eμ∓ 3.5 × 10−4[7] jCμeeHj; jCeμeHj < 2.5 × 10−4 h→ τμ∓ 1.5 × 10−2[8] jCμτeHj; jCτμeHj < 1.6 × 10−3 h→ eτ∓ 6.9 × 10−3[7] jCeτeHj; jCτeeHj < 1.1 × 10−3 τ → ee¯e 2.7 × 10−8[9] jCeτee
ll þ Ceeeτll þ geL½CHlð1Þeτ þ CeτHlð3Þ − δCeτpenguinj < 2.8 × 10−4
jCeτee
le þ geR½CeτHlð1Þþ CeτHlð3Þ − δCeτpenguinj < 4.0 × 10−4
τ → eμ¯μ 2.7 × 10−8[9] jCeτμμ
ll þ Cμμeτll þ Cellμμτþ Cμτeμll þ geL½CeHlð1Þτ þ CeHlð3Þτ − δCepenguinτ j < 4.0 × 10−4
jCeτμμle þ ge
R½CμτHlð1Þþ CeHτlð3Þ − δCepenguinτ j < 4.0 × 10−4
τ → μe¯e 1.8 × 10−8[9] jCμτee
ll þ Ceeμτll Cμeeτll þ Ceτμell þ gLe½CμτHlð1Þþ CμτHlð3Þ − δCμτpenguinj < 3.2 × 10−4
jCμτeele þ ge R½C μτ Hlð1Þþ CμτHlð3Þ − δCμτpenguinj < 3.2 × 10−4 τ → μμ¯μ 2.1 × 10−8[9] jCμτμμ ll þ Cμμμτll þ geL½CHlð1Þμτ þ CμτHlð3Þ − δCμτpenguinj < 2.5 × 10−4 jCμτμμle þ ge R½CμτHlð1Þþ CμτHlð3Þ − δCμτpenguinj < 3.5 × 10−4 τ → ee¯μ 1.5 × 10−8[9] jCeτeμ ll þ Ceμeτll j < 3.2 × 10−4 τ → μμ¯e 1.7 × 10−8[9] jCμτμe ll þ Cμeμτll j < 3.2 × 10−4 μ → 3e 1 × 10−12 [10] jCeμee ll þ Ceeeμll þ geL½C eμ
Hlð1Þþ CeμHlð3Þ − δCeμpenguinj < 7.1 × 10−7
jCeμeele þ ge R½C
eμ
Hlð1Þþ CeμHlð3Þ − δCeμpenguinj < 1.0 × 10−6
1. Rates and calculations a.Z → lαlβ decay
When the Higgs gets a vev, the “penguin” operators OHlð1ÞandOHlð3Þgenerate a vertex involving the Z and two
charged leptons. If the flavor-changing Z-fermion vertex is written in a SM-like form :−lαZμ g2c
WγμðgV− gAγ5Þlβ, then
gV ¼ gA¼ −ðCHlð1Þþ CHlð3ÞÞ
v2
Λ2 ðE1Þ
(for v∼ mt).
The branching ratio can be written BRðZ → lαlβÞ ¼2.5 GeVMZ g
2
48πc2 W
ðjgVj2þ jgAj2Þ ðE2Þ
where 2.5 GeV is the Z width in the SM. Since OHlð1Þ and OHlð3Þ are Hermitian, the conjugate process
Z→ lβlα necessarily occurs at the same rate, so the BR to the experimental final state is
BRðZ → l αl∓βÞ ¼ BRðZ → lαlβÞ þ BRðZ → lβlαÞ ¼ MZ 2.5 GeV g2 12πc2 W ðCαβHlð1Þþ CαβHlð3Þ 2v4 Λ4 ðE3Þ and the bounds we obtain on the operator coefficients, evaluated at∼mW, are given in TableI.
b.h → l+
αeβ−, eα+lβ− decays
The flavor-changing Higgs decays occur via the non-Hermitian operator OeH. When the Higgs has a vev, it
induces the Feynman rules for a flavor-changing Higgs vertex with two fermions:
CαβeHOαβeH→i3C αβ eHv2 ffiffiffi 2 p Λ2 PR; CβαeHO βα eH→i 3CβαeHffiffiffi v2 2 p Λ2 PL: ðE4Þ
We calculate the flavor-changing branching ratio by compar-ing to BRðh → bbÞ ¼ 0.575 0.32 (from the Appendix of the Higgs Working Group Report[36], for mh¼ 125.1 GeV),
assuming the Feynman rule for hbb is−piffiffi2ybðmhÞPL;R. We
use a one-loop approximation[15]for the running b mass ybðmhÞv ¼ mbðmbÞ αðmhÞ αðmbÞ γð0Þ m=2βð0Þ ≃ 3.0 GeV ðE5Þ where αðmhÞ ≃ 0.12, αðmbÞ ≃ 0.23, γð0Þm ¼ 8, βð0Þ¼ 23=3 and mbðmbÞ ¼ 4.2 GeV.
The operator OeH is not hermitian, but is always included in the Lagrangianþ H:c:. So CeμeHOeμeHþ H:c: will induce both h→ eLμR and h→ μReL at the same rate:
BRðh → eLμRÞ BRðh → bbÞ ¼ 9 jCeμeHj2v4 6y2 bΛ4 ; ðE6Þ
where downstairs there is a 3 for quark color sums, and a 2 from the chiral projectors in the lepton decay. The exper-imental search sums the eLμRandμReLfinal states, so we
obtain 3v4jCαβeHj2 Λ4 ; 3v4 jCβαeHj2 Λ4 ≤ y2bðmhÞ BRðh → l αl∓βÞ BRðh → bbÞ ðE7Þ and the resulting constraints are given in TableI.
c. Including the low energy decays
The flavor-changing τ and μ decays listed in Table I
occur at energies∼mμ, mτ, so the decay rates are usually written in terms of the coefficients of dimension-six operators from the QCD × QED invariant basis appropriate at low energies. These“low energy” coefficients, which we denote with a tilde ˜C, can be expressed in terms of SMEFT coefficients at mW by running them up to mW, then matching the QCD × QED-invariant operator basis onto the SMEFT. This was performed in[34]forμ → eγ, so we use the results of[34]for the radiative decays of Sec.E 1 e. Reference [35] studied the renormalization group evolu-tion, below the weak scale, of the coefficients who mediate μ → eee (as well those for as μ → eγ and μ → e con-version); we use these results, combined with the weak-scale matching conditions of [34], for the discussion in Sec. E 1 d of three body leptonic decays ofτs and μs. The minor differences between μ and τ decays are discussed in Sec.E 1 d.
In the EFT below mW, we use the basis of
lepton-flavor-changing four-fermion operators introduced in [2,34] for μ ↔ e flavor change.7
The operators and coefficients have as subscript their Lorentz structure (V, S, T) and the chiral projection operators of the two fermion bilinears, and the flavor indices of the four fermions as superscript. They wear tildes to distinguish them from the coefficients of SMEFT operators. We restrict to the dipole and vector operators, and neglect the scalars and tensors, which will turn out to be irrelevant for our study of LFV operators generated by double-insertions of LNV operators. So the four-fermion operator basis below mW is
δL4f¼ X αβ X f ½ ˜CαβffV;LLðeαγωPLeβÞðfγωPLfÞ þ ˜CαβffV;LRðeαγωP LeβÞðfγωPRfÞ þ H:c: þX αβσρ ½ ˜CαβσρV;LLðeαγωPLeβÞðeσγωPLeρÞ þ H:c: ðE8Þ
7In this basis, the flavor indices are written explicitly, so the 2
